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Mirrors > Home > MPE Home > Th. List > elovmpo | Structured version Visualization version GIF version |
Description: Utility lemma for
two-parameter classes.
EDITORIAL: can simplify isghm 19016, islmhm 20532. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
elovmpo.d | ⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) |
elovmpo.c | ⊢ 𝐶 ∈ V |
elovmpo.e | ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
elovmpo | ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elovmpo.d | . . . 4 ⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | elmpocl 7599 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) |
3 | elovmpo.c | . . . . . . 7 ⊢ 𝐶 ∈ V | |
4 | 3 | gen2 1799 | . . . . . 6 ⊢ ∀𝑎∀𝑏 𝐶 ∈ V |
5 | elovmpo.e | . . . . . . . 8 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸) | |
6 | 5 | eleq1d 2819 | . . . . . . 7 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V)) |
7 | 6 | spc2gv 3561 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑎∀𝑏 𝐶 ∈ V → 𝐸 ∈ V)) |
8 | 4, 7 | mpi 20 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝐸 ∈ V) |
9 | 5, 1 | ovmpoga 7513 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸) |
10 | 8, 9 | mpd3an3 1463 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐷𝑌) = 𝐸) |
11 | 10 | eleq2d 2820 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹 ∈ 𝐸)) |
12 | 2, 11 | biadanii 821 | . 2 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸)) |
13 | df-3an 1090 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ 𝐸)) | |
14 | 12, 13 | bitr4i 278 | 1 ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 Vcvv 3447 (class class class)co 7361 ∈ cmpo 7363 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 |
This theorem is referenced by: isgim 19060 oppglsm 19432 islmim 20567 |
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