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 Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1
caofref.2
caofcom.3
Assertion
Ref Expression
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5
21ffvelrnda 5862 . . . 4
3 caofcom.3 . . . . 5
43ffvelrnda 5862 . . . 4
5 caofrss.4 . . . . . 6
65ralrimivva 2790 . . . . 5
76adantr 452 . . . 4
8 breq1 4207 . . . . . 6
9 breq1 4207 . . . . . 6
108, 9imbi12d 312 . . . . 5
11 breq2 4208 . . . . . 6
12 breq2 4208 . . . . . 6
1311, 12imbi12d 312 . . . . 5
1410, 13rspc2va 3051 . . . 4
152, 4, 7, 14syl21anc 1183 . . 3
1615ralimdva 2776 . 2
17 ffn 5583 . . . 4
181, 17syl 16 . . 3
19 ffn 5583 . . . 4
203, 19syl 16 . . 3
21 caofref.1 . . 3
22 inidm 3542 . . 3
23 eqidd 2436 . . 3
24 eqidd 2436 . . 3
2518, 20, 21, 21, 22, 23, 24ofrfval 6305 . 2
2618, 20, 21, 21, 22, 23, 24ofrfval 6305 . 2
2716, 25, 263imtr4d 260 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697   class class class wbr 4204   wfn 5441  wf 5442  cfv 5446   cofr 6296 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ofr 6298
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