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Mirrors > Home > MPE Home > Th. List > elmapresaun | Structured version Visualization version GIF version |
Description: fresaun 6542 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
elmapresaun | ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8417 | . . 3 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐹:𝐴⟶𝐶) | |
2 | elmapi 8417 | . . 3 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → 𝐺:𝐵⟶𝐶) | |
3 | id 22 | . . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) | |
4 | fresaun 6542 | . . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | |
5 | 1, 2, 3, 4 | syl3an 1152 | . 2 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
6 | elmapex 8416 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V)) | |
7 | 6 | simpld 495 | . . . 4 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐶 ∈ V) |
8 | 7 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐶 ∈ V) |
9 | 6 | simprd 496 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐴 ∈ V) |
10 | elmapex 8416 | . . . . . 6 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V)) | |
11 | 10 | simprd 496 | . . . . 5 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → 𝐵 ∈ V) |
12 | unexg 7461 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
13 | 9, 11, 12 | syl2an 595 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵)) → (𝐴 ∪ 𝐵) ∈ V) |
14 | 13 | 3adant3 1124 | . . 3 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐴 ∪ 𝐵) ∈ V) |
15 | 8, 14 | elmapd 8409 | . 2 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ↔ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)) |
16 | 5, 15 | mpbird 258 | 1 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ↾ cres 5550 ⟶wf 6344 (class class class)co 7145 ↑m cmap 8395 |
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 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
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-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 |
This theorem is referenced by: satfv1lem 32506 diophin 39247 eldioph4b 39286 diophren 39288 |
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