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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 3572 for operation values, analogous to rspceov 7315. (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 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 44626 | . . 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 44626 | . . 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)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ((caov 44566 |
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 5222 ax-nul 5229 ax-pr 5351 |
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 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-br 5075 df-opab 5137 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-res 5597 df-iota 6385 df-fun 6429 df-fv 6435 df-aiota 44533 df-dfat 44567 df-afv 44568 df-aov 44569 |
This theorem is referenced by: (None) |
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