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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 3632 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 7152 | . . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦)) | |
2 | 1 | eqeq2d 2829 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦))) |
3 | oveq2 7153 | . . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷)) | |
4 | 3 | eqeq2d 2829 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷))) |
5 | 2, 4 | rspc2ev 3632 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: iunfictbso 9528 genpprecl 10411 elz2 11987 zaddcl 12010 znq 12340 qaddcl 12352 qmulcl 12354 qreccl 12356 xpsff1o 16828 mndpfo 17922 gafo 18364 lsmelvalix 18695 lsmelvalmi 18706 evthicc2 23988 i1fadd 24223 i1fmul 24224 2clwwlk2clwwlk 28056 isgrpoi 28202 shscli 29021 shsva 29024 shunssi 29072 pjpjhth 29129 spanunsni 29283 pjjsi 29404 ofrn2 30315 pstmfval 31035 ismblfin 34814 itg2addnc 34827 blbnd 34946 isgrpda 35114 sbgoldbalt 43823 |
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