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Mirrors > Home > MPE Home > Th. List > Mathboxes > mof0ALT | Structured version Visualization version GIF version |
Description: Alternate proof for mof0 47890 with stronger requirements on distinct variables. Uses mo4 2556. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mof0ALT | ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f00 6779 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
2 | 1 | simplbi 497 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
3 | f00 6779 | . . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simplbi 497 | . . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅) |
5 | eqtr3 2754 | . . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔) | |
6 | 2, 4, 5 | syl2an 595 | . . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔) |
7 | 6 | gen2 1791 | . 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔) |
8 | feq1 6703 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅)) | |
9 | 8 | mo4 2556 | . 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)) |
10 | 7, 9 | mpbir 230 | 1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1532 = wceq 1534 ∃*wmo 2528 ∅c0 4323 ⟶wf 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: (None) |
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