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Mirrors > Home > MPE Home > Th. List > fco3OLD | Structured version Visualization version GIF version |
Description: Obsolete version of funcofd 6556 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fco3OLD.1 | ⊢ (𝜑 → Fun 𝐹) |
fco3OLD.2 | ⊢ (𝜑 → Fun 𝐺) |
Ref | Expression |
---|---|
fco3OLD | ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco3OLD.1 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
2 | fco3OLD.2 | . . . . 5 ⊢ (𝜑 → Fun 𝐺) | |
3 | funco 6398 | . . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2anc 587 | . . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
5 | fdmrn 6555 | . . . 4 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺)) | |
6 | 4, 5 | sylib 221 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺)) |
7 | dmco 6098 | . . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹) | |
8 | 7 | feq2i 6515 | . . 3 ⊢ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺)) |
9 | 6, 8 | sylib 221 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺)) |
10 | rncoss 5826 | . . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹) |
12 | 9, 11 | fssd 6541 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 ∘ ccom 5540 Fun wfun 6352 ⟶wf 6354 |
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-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: (None) |
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