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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 3637 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 7165 | . . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦)) | |
2 | 1 | eqeq2d 2834 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦))) |
3 | oveq2 7166 | . . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷)) | |
4 | 3 | eqeq2d 2834 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷))) |
5 | 2, 4 | 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 (class class class)co 7158 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 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-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: iunfictbso 9542 genpprecl 10425 elz2 12002 zaddcl 12025 znq 12355 qaddcl 12367 qmulcl 12369 qreccl 12371 xpsff1o 16842 mndpfo 17936 gafo 18428 lsmelvalix 18768 lsmelvalmi 18779 evthicc2 24063 i1fadd 24298 i1fmul 24299 2clwwlk2clwwlk 28131 isgrpoi 28277 shscli 29096 shsva 29099 shunssi 29147 pjpjhth 29204 spanunsni 29358 pjjsi 29479 ofrn2 30389 pstmfval 31138 ismblfin 34935 itg2addnc 34948 blbnd 35067 isgrpda 35235 sbgoldbalt 43953 |
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