Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rspceov | Structured version Visualization version GIF version |
Description: A frequently used special case of rspc2ev 3539 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 7198 | . . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦)) | |
2 | 1 | eqeq2d 2747 | . 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦))) |
3 | oveq2 7199 | . . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷)) | |
4 | 3 | eqeq2d 2747 | . 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷))) |
5 | 2, 4 | rspc2ev 3539 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 |
This theorem is referenced by: iunfictbso 9693 genpprecl 10580 elz2 12159 zaddcl 12182 znq 12513 qaddcl 12526 qmulcl 12528 qreccl 12530 xpsff1o 17026 mndpfo 18150 gafo 18644 lsmelvalix 18984 lsmelvalmi 18995 evthicc2 24311 i1fadd 24546 i1fmul 24547 2clwwlk2clwwlk 28387 isgrpoi 28533 shscli 29352 shsva 29355 shunssi 29403 pjpjhth 29460 spanunsni 29614 pjjsi 29735 ofrn2 30650 elringlsmd 31250 pstmfval 31514 ismblfin 35504 itg2addnc 35517 blbnd 35631 isgrpda 35799 sbgoldbalt 44849 |
Copyright terms: Public domain | W3C validator |