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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 3595 for operation values, analogous to rspceov 7409. (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 2738 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐹 = 𝐹) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶) | |
3 | eqidd 2738 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝑦 = 𝑦) | |
4 | 1, 2, 3 | aoveq123d 45484 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) ) |
5 | 4 | eqeq2d 2748 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) )) |
6 | eqidd 2738 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐹 = 𝐹) | |
7 | eqidd 2738 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝐶 = 𝐶) | |
8 | id 22 | . . . 4 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷) | |
9 | 6, 7, 8 | aoveq123d 45484 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) ) |
10 | 9 | eqeq2d 2748 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) )) |
11 | 5, 10 | rspc2ev 3595 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ((caov 45424 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-res 5650 df-iota 6453 df-fun 6503 df-fv 6509 df-aiota 45391 df-dfat 45425 df-afv 45426 df-aov 45427 |
This theorem is referenced by: (None) |
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