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Mirrors > Home > ILE Home > Th. List > foun | GIF version |
Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
foun | ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5355 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 5355 | . . . 4 ⊢ (𝐺:𝐶–onto→𝐷 → 𝐺 Fn 𝐶) | |
3 | 1, 2 | anim12i 336 | . . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶)) |
4 | fnun 5237 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)) | |
5 | 3, 4 | sylan 281 | . 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)) |
6 | rnun 4955 | . . 3 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺) | |
7 | forn 5356 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | ad2antrr 480 | . . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐹 = 𝐵) |
9 | forn 5356 | . . . . 5 ⊢ (𝐺:𝐶–onto→𝐷 → ran 𝐺 = 𝐷) | |
10 | 9 | ad2antlr 481 | . . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐺 = 𝐷) |
11 | 8, 10 | uneq12d 3236 | . . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵 ∪ 𝐷)) |
12 | 6, 11 | syl5eq 2185 | . 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷)) |
13 | df-fo 5137 | . 2 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷))) | |
14 | 5, 12, 13 | sylanbrc 414 | 1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 ran crn 4548 Fn wfn 5126 –onto→wfo 5129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 |
This theorem is referenced by: (None) |
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