Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoresf1b | Structured version Visualization version GIF version |
Description: A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.) |
Ref | Expression |
---|---|
fcores.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fcores.e | ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) |
fcores.p | ⊢ 𝑃 = (◡𝐹 “ 𝐶) |
fcores.x | ⊢ 𝑋 = (𝐹 ↾ 𝑃) |
fcores.g | ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) |
fcores.y | ⊢ 𝑌 = (𝐺 ↾ 𝐸) |
Ref | Expression |
---|---|
fcoresf1b | ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcores.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐹:𝐴⟶𝐵) |
3 | fcores.e | . . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
4 | fcores.p | . . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
5 | fcores.x | . . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
6 | fcores.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → 𝐺:𝐶⟶𝐷) |
8 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
9 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) | |
10 | 2, 3, 4, 5, 7, 8, 9 | fcoresf1 44541 | . . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–1-1→𝐷) → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)) |
11 | 10 | ex 413 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))) |
12 | f1co 6674 | . . . 4 ⊢ ((𝑌:𝐸–1-1→𝐷 ∧ 𝑋:𝑃–1-1→𝐸) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷) | |
13 | 12 | ancoms 459 | . . 3 ⊢ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝑌 ∘ 𝑋):𝑃–1-1→𝐷) |
14 | 1, 3, 4, 5, 6, 8 | fcores 44539 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) |
15 | f1eq1 6657 | . . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–1-1→𝐷)) |
17 | 13, 16 | syl5ibr 245 | . 2 ⊢ (𝜑 → ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)) |
18 | 11, 17 | impbid 211 | 1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∩ cin 3885 ◡ccnv 5583 ran crn 5585 ↾ cres 5586 “ cima 5587 ∘ ccom 5588 ⟶wf 6422 –1-1→wf1 6423 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-fv 6434 |
This theorem is referenced by: fcoresf1ob 44545 |
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