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Mirrors > Home > MPE Home > Th. List > rspceov | Structured version Visualization version GIF version |
Description: A frequently used special case of rspc2ev 3623 for operation values. (Contributed by NM, 21-Mar-2007.) |
Ref | Expression |
---|---|
rspceov | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7411 | . . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦)) | |
2 | 1 | eqeq2d 2744 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦))) |
3 | oveq2 7412 | . . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷)) | |
4 | 3 | eqeq2d 2744 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷))) |
5 | 2, 4 | rspc2ev 3623 | 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 3071 (class class class)co 7404 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7407 |
This theorem is referenced by: iunfictbso 10105 genpprecl 10992 elz2 12572 zaddcl 12598 znq 12932 qaddcl 12945 qmulcl 12947 qreccl 12949 xpsff1o 17509 mndpfo 18644 gafo 19154 lsmelvalix 19502 lsmelvalmi 19513 evthicc2 24959 i1fadd 25194 i1fmul 25195 2clwwlk2clwwlk 29583 isgrpoi 29729 shscli 30548 shsva 30551 shunssi 30599 pjpjhth 30656 spanunsni 30810 pjjsi 30931 ofrn2 31843 elringlsmd 32469 pstmfval 32814 ismblfin 36467 itg2addnc 36480 blbnd 36593 isgrpda 36761 sbgoldbalt 46384 |
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