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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 3583 for operation values, analogous to rspceov 7182. (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 2799 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶) | |
3 | eqidd 2799 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦) | |
4 | 1, 2, 3 | aoveq123d 43734 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) ) |
5 | 4 | eqeq2d 2809 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) )) |
6 | eqidd 2799 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹) | |
7 | eqidd 2799 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶) | |
8 | id 22 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷) | |
9 | 6, 7, 8 | aoveq123d 43734 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) ) |
10 | 9 | eqeq2d 2809 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) )) |
11 | 5, 10 | rspc2ev 3583 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ((caov 43674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-aiota 43642 df-dfat 43675 df-afv 43676 df-aov 43677 |
This theorem is referenced by: (None) |
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