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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1386 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1386.1 | ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) |
| bnj1386.2 | ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) |
| bnj1386.3 | ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) |
| bnj1386.4 | ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| bnj1386 | ⊢ (𝜓 → Fun ∪ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1386.1 | . 2 ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) | |
| 2 | bnj1386.2 | . 2 ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) | |
| 3 | bnj1386.3 | . 2 ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) | |
| 4 | bnj1386.4 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴) | |
| 5 | biid 261 | . 2 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ ∈ 𝐴 Fun ℎ) | |
| 6 | eqid 2737 | . 2 ⊢ (dom ℎ ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔) | |
| 7 | biid 261 | . 2 ⊢ ((∀ℎ ∈ 𝐴 Fun ℎ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ (dom ℎ ∩ dom 𝑔)) = (𝑔 ↾ (dom ℎ ∩ dom 𝑔))) ↔ (∀ℎ ∈ 𝐴 Fun ℎ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ (dom ℎ ∩ dom 𝑔)) = (𝑔 ↾ (dom ℎ ∩ dom 𝑔)))) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | bnj1385 34846 | 1 ⊢ (𝜓 → Fun ∪ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ∪ cuni 4907 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: bnj1384 35046 |
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