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Mirrors > Home > MPE Home > Th. List > Mathboxes > k0004lem1 | Structured version Visualization version GIF version |
Description: Application of ssin 4194 to range of a function. (Contributed by RP, 1-Apr-2021.) |
Ref | Expression |
---|---|
k0004lem1 | ⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnima 6635 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) | |
2 | 1 | sseq1d 3979 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝐹 “ 𝐴) ⊆ 𝐶 ↔ ran 𝐹 ⊆ 𝐶)) |
3 | 2 | anbi2d 630 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶))) |
4 | ssin 4194 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)) | |
5 | 3, 4 | bitrdi 287 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))) |
6 | 5 | pm5.32i 576 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))) |
7 | df-f 6504 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
8 | 7 | anbi1i 625 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶)) |
9 | anass 470 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶))) | |
10 | 8, 9 | bitri 275 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶))) |
11 | df-f 6504 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))) | |
12 | 6, 10, 11 | 3bitr4i 303 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶)) |
13 | feq3 6655 | . 2 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → (𝐹:𝐴⟶𝐷 ↔ 𝐹:𝐴⟶(𝐵 ∩ 𝐶))) | |
14 | 12, 13 | bitr4id 290 | 1 ⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∩ cin 3913 ⊆ wss 3914 ran crn 5638 “ cima 5640 Fn wfn 6495 ⟶wf 6496 |
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-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-fun 6502 df-fn 6503 df-f 6504 |
This theorem is referenced by: k0004lem2 42512 |
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