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Mirrors > Home > MPE Home > Th. List > fncoOLD | Structured version Visualization version GIF version |
Description: Obsolete version of fnco 6472 as of 20-Sep-2024. (Contributed by NM, 22-May-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fncoOLD | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6457 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | fnfun 6457 | . . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
3 | funco 6398 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2an 599 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺)) |
5 | 4 | 3adant3 1134 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → Fun (𝐹 ∘ 𝐺)) |
6 | fndm 6459 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | sseq2d 3919 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ 𝐴)) |
8 | 7 | biimpar 481 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → ran 𝐺 ⊆ dom 𝐹) |
9 | dmcosseq 5827 | . . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
11 | 10 | 3adant2 1133 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
12 | fndm 6459 | . . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
13 | 12 | 3ad2ant2 1136 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵) |
14 | 11, 13 | eqtrd 2771 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = 𝐵) |
15 | df-fn 6361 | . 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵)) | |
16 | 5, 14, 15 | sylanbrc 586 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ⊆ wss 3853 dom cdm 5536 ran crn 5537 ∘ ccom 5540 Fun wfun 6352 Fn wfn 6353 |
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-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
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-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 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-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 |
This theorem is referenced by: (None) |
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