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Mirrors > Home > MPE Home > Th. List > dffun2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of dffun2 6553 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dffun2OLD | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 6545 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
2 | cotrg 6108 | . . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧)) | |
3 | alcom 2156 | . . . . 5 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧)) | |
4 | vex 3478 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
5 | vex 3478 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | brcnv 5882 | . . . . . . . 8 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
7 | 6 | anbi1i 624 | . . . . . . 7 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)) |
8 | vex 3478 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
9 | 8 | ideq 5852 | . . . . . . 7 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧) |
10 | 7, 9 | imbi12i 350 | . . . . . 6 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) |
11 | 10 | 3albii 1823 | . . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) |
12 | 3, 11 | bitri 274 | . . . 4 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) |
13 | 2, 12 | bitri 274 | . . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) |
14 | 13 | anbi2i 623 | . 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) |
15 | 1, 14 | bitri 274 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ⊆ wss 3948 class class class wbr 5148 I cid 5573 ◡ccnv 5675 ∘ ccom 5680 Rel wrel 5681 Fun wfun 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6545 |
This theorem is referenced by: (None) |
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