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| Mirrors > Home > MPE Home > Th. List > fnco | Structured version Visualization version GIF version | ||
| Description: Composition of two functions with domains as a function with domain. (Contributed by NM, 22-May-2006.) (Proof shortened by AV, 20-Sep-2024.) |
| Ref | Expression |
|---|---|
| fnco | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnfun 6668 | . . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
| 2 | fncofn 6685 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) | |
| 3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) |
| 4 | 3 | 3adant3 1133 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) |
| 5 | cnvimassrndm 6172 | . . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (◡𝐺 “ 𝐴) = dom 𝐺) | |
| 6 | 5 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (◡𝐺 “ 𝐴) = dom 𝐺) |
| 7 | fndm 6671 | . . . . 5 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
| 8 | 7 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵) |
| 9 | 6, 8 | eqtr2d 2778 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → 𝐵 = (◡𝐺 “ 𝐴)) |
| 10 | 9 | fneq2d 6662 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))) |
| 11 | 4, 10 | mpbird 257 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ∘ ccom 5689 Fun wfun 6555 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnfco 6773 fsplitfpar 8143 fipreima 9398 updjudhcoinlf 9972 updjudhcoinrg 9973 cshco 14875 swrdco 14876 isofn 17819 prdsinvlem 19067 prdsmgp 20148 pws1 20322 frlmbas 21775 frlmup3 21820 frlmup4 21821 evlslem1 22106 upxp 23631 uptx 23633 0vfval 30625 xppreima2 32661 psgnfzto1stlem 33120 tocycfvres1 33130 tocycfvres2 33131 cycpmfvlem 33132 cycpmfv3 33135 cycpmco2 33153 sseqfv1 34391 sseqfn 34392 sseqfv2 34396 volsupnfl 37672 ftc1anclem5 37704 ftc1anclem8 37707 choicefi 45205 fourierdlem42 46164 fcoreslem4 47078 ackvalsucsucval 48609 |
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