Theorem unir1 4677

Description: The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281.

Assertion

Ref	Expression
unir1	⊢ ∪(R1 “ On) = V

Proof of Theorem unir1

Step	Hyp	Ref	Expression
1		r1fnon 4660	. . . . . 6 ⊢ R1 Fn On
2		fndm 3593	. . . . . 6 ⊢ (R1 Fn On → dom R1 = On)
3	1, 2	ax-mp 7	. . . . 5 ⊢ dom R1 = On
4		imaeq2 3408	. . . . 5 ⊢ (dom R1 = On → (R1 “ dom R1) = (R1 “ On))
5	3, 4	ax-mp 7	. . . 4 ⊢ (R1 “ dom R1) = (R1 “ On)
6		imadmrn 3420	. . . 4 ⊢ (R1 “ dom R1) = ran R1
7	5, 6	eqtr3 1500	. . 3 ⊢ (R1 “ On) = ran R1
8	7	unieqi 2515	. 2 ⊢ ∪(R1 “ On) = ∪ran R1
9		fniunfv 3871	. . 3 ⊢ (R1 Fn On → ∪x ∈ On (R1 ‘x) = ∪ran R1)
10	1, 9	ax-mp 7	. 2 ⊢ ∪x ∈ On (R1 ‘x) = ∪ran R1
11		jech9.3 4676	. 2 ⊢ ∪x ∈ On (R1 ‘x) = V
12	8, 10, 11	3eqtr2 1504	1 ⊢ ∪(R1 “ On) = V

Syntax hints:   = wceq 958  Vcvv 1814  ∪cuni 2507  ∪ciun 2570  Oncon0 2954  dom cdm 3176  ran crn 3177   “ cima 3179   Fn wfn 3183   ‘cfv 3188  R₁cr1 4651

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653

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