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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspceaov | Structured version Visualization version GIF version |
Description: A frequently used special case of rspc2ev 3637 for operation values, analogous to rspceov 7205. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
rspceaov | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2824 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶) | |
3 | eqidd 2824 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦) | |
4 | 1, 2, 3 | aoveq123d 43384 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) ) |
5 | 4 | eqeq2d 2834 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) )) |
6 | eqidd 2824 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹) | |
7 | eqidd 2824 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶) | |
8 | id 22 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷) | |
9 | 6, 7, 8 | aoveq123d 43384 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) ) |
10 | 9 | eqeq2d 2834 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) )) |
11 | 5, 10 | rspc2ev 3637 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ((caov 43324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-aiota 43292 df-dfat 43325 df-afv 43326 df-aov 43327 |
This theorem is referenced by: (None) |
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