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Mirrors > Home > MPE Home > Th. List > foun | Structured version Visualization version 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 6759 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 6759 | . . . 4 ⊢ (𝐺:𝐶–onto→𝐷 → 𝐺 Fn 𝐶) | |
3 | 1, 2 | anim12i 614 | . . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶)) |
4 | fnun 6615 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)) | |
5 | 3, 4 | sylan 581 | . 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)) |
6 | rnun 6099 | . . 3 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺) | |
7 | forn 6760 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | ad2antrr 725 | . . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐹 = 𝐵) |
9 | forn 6760 | . . . . 5 ⊢ (𝐺:𝐶–onto→𝐷 → ran 𝐺 = 𝐷) | |
10 | 9 | ad2antlr 726 | . . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐺 = 𝐷) |
11 | 8, 10 | uneq12d 4125 | . . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵 ∪ 𝐷)) |
12 | 6, 11 | eqtrid 2789 | . 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷)) |
13 | df-fo 6503 | . 2 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷))) | |
14 | 5, 12, 13 | sylanbrc 584 | 1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 ran crn 5635 Fn wfn 6492 –onto→wfo 6495 |
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-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: (None) |
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