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Mirrors > Home > MPE Home > Th. List > f1od | Structured version Visualization version GIF version |
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.) |
Ref | Expression |
---|---|
f1od.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
f1od.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
f1od.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) |
f1od.4 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
f1od | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
2 | f1od.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
3 | f1od.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) | |
4 | f1od.4 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
5 | 1, 2, 3, 4 | f1ocnvd 6884 | . 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
6 | 5 | simpld 475 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ◡ccnv 5113 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: cnvf1o 7276 ixpsnf1o 7948 en2d 7991 pw2f1o 8065 seqf1olem1 12840 resf1o 29505 fpwrelmap 29508 actfunsnf1o 30682 |
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