Theorem rspceaov 44689

Description: A frequently used special case of rspc2ev 3572 for operation values, analogous to rspceov 7322. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
rspceaov	⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐷(𝑥)

Proof of Theorem rspceaov

Step	Hyp	Ref	Expression
1		eqidd 2739	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹)
2		id 22	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶)
3		eqidd 2739	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦)
4	1, 2, 3	aoveq123d 44670	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
5	4	eqeq2d 2749	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6		eqidd 2739	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹)
7		eqidd 2739	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶)
8		id 22	. . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷)
9	6, 7, 8	aoveq123d 44670	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
10	9	eqeq2d 2749	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
11	5, 10	rspc2ev 3572	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ((caov 44610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612  df-aov 44613

This theorem is referenced by: (None)