Theorem ssoprab2i 5581

Description: Inference of operation class abstraction subclass from implication. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 11-Nov-1995.) (Revised by set.mm contributors, 24-Jul-2012.)

Hypothesis

Ref	Expression
ssoprab2i.1	⊢ (φ → ψ)

Assertion

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

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem ssoprab2i

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssoprab2i.1	. . . . 5 ⊢ (φ → ψ)
2	1	anim2i 552	. . . 4 ⊢ ((w = ⟨x, y⟩ ∧ φ) → (w = ⟨x, y⟩ ∧ ψ))
3	2	2eximi 1577	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) → ∃x∃y(w = ⟨x, y⟩ ∧ ψ))
4	3	ssopab2i 4715	. 2 ⊢ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} ⊆ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ψ)}
5		dfoprab2 5559	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
6		dfoprab2 5559	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ψ)}
7	4, 5, 6	3sstr4i 3311	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} ⊆ {⟨⟨x, y⟩, z⟩ ∣ ψ}

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ⊆ wss 3258  ⟨cop 4562  {copab 4623  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-oprab 5529

This theorem is referenced by: (None)