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Mirrors > Home > MPE Home > Th. List > mpofunOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mpofun 7535 as of 23-Jul-2024. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mpofun.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
mpofunOLD | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2757 | . . . . . 6 ⊢ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐶) → 𝑧 = 𝑤) | |
2 | 1 | ad2ant2l 743 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤) |
3 | 2 | gen2 1797 | . . . 4 ⊢ ∀𝑧∀𝑤((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤) |
4 | eqeq1 2735 | . . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝐶 ↔ 𝑤 = 𝐶)) | |
5 | 4 | anbi2d 628 | . . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))) |
6 | 5 | mo4 2559 | . . . 4 ⊢ (∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧∀𝑤((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)) |
7 | 3, 6 | mpbir 230 | . . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) |
8 | 7 | funoprab 7533 | . 2 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
9 | mpofun.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
10 | df-mpo 7417 | . . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
11 | 9, 10 | eqtri 2759 | . . 3 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
12 | 11 | funeqi 6569 | . 2 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
13 | 8, 12 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ∃*wmo 2531 Fun wfun 6537 {coprab 7413 ∈ cmpo 7414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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 df-oprab 7416 df-mpo 7417 |
This theorem is referenced by: (None) |
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