Theorem ssoprab2i 4014

Description: Inference of operation class abstraction subclass from implication.

Hypothesis

Ref	Expression
ssoprab2i.1	⊢ (φ → ψ)

Assertion

Ref	Expression
ssoprab2i	⊢ {⟨⟨x, y⟩, z⟩∣φ} ⊆ {⟨⟨x, y⟩, z⟩∣ψ}

Distinct variable group:   x,y,z

Proof of Theorem ssoprab2i

Step	Hyp	Ref	Expression
1		ssoprab2i.1	. . . . 5 ⊢ (φ → ψ)
2	1	anim2i 335	. . . 4 ⊢ ((w = ⟨x, y⟩ ⋀ φ) → (w = ⟨x, y⟩ ⋀ ψ))
3	2	19.22i2 1043	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ⋀ φ) → ∃x∃y(w = ⟨x, y⟩ ⋀ ψ))
4	3	ssopab2i 2829	. 2 ⊢ {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ⋀ φ)} ⊆ {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ⋀ ψ)}
5		dfoprab2 3997	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ⋀ φ)}
6		dfoprab2 3997	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣ψ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ⋀ ψ)}
7	4, 5, 6	3sstr4 2103	1 ⊢ {⟨⟨x, y⟩, z⟩∣φ} ⊆ {⟨⟨x, y⟩, z⟩∣ψ}

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958  ∃wex 982   ⊆ wss 2050  ⟨cop 2415  {copab 2671  {copab2 3970

This theorem is referenced by:  blfval 7832  nvvcop 8209

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-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-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-oprab 3972

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