Theorem rspceaov 47954

Description: A frequently used special case of rspc2ev 3594 for operation values, analogous to rspceov 7459. (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 2764	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹)
2		id 23	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶)
3		eqidd 2764	. . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦)
4	1, 2, 3	aoveq123d 47935	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
5	4	eqeq2d 2774	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6		eqidd 2764	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹)
7		eqidd 2764	. . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶)
8		id 23	. . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷)
9	6, 7, 8	aoveq123d 47935	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
10	9	eqeq2d 2774	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
11	5, 10	rspc2ev 3594	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ((caov 47875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-aiota 47842  df-dfat 47876  df-afv 47877  df-aov 47878

This theorem is referenced by: (None)