Theorem rspceaov 47214

Description: A frequently used special case of rspc2ev 3634 for operation values, analogous to rspceov 7481. (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 2737	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹)
2		id 22	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶)
3		eqidd 2737	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦)
4	1, 2, 3	aoveq123d 47195	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
5	4	eqeq2d 2747	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6		eqidd 2737	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹)
7		eqidd 2737	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶)
8		id 22	. . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷)
9	6, 7, 8	aoveq123d 47195	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
10	9	eqeq2d 2747	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
11	5, 10	rspc2ev 3634	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∃wrex 3069   ((caov 47135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-aiota 47102  df-dfat 47136  df-afv 47137  df-aov 47138

This theorem is referenced by: (None)