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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 262 | . 2 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ ∈ 𝐴 Fun ℎ) | |
6 | eqid 2818 | . 2 ⊢ (dom ℎ ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔) | |
7 | biid 262 | . 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 32003 | 1 ⊢ (𝜓 → Fun ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ∪ cuni 4830 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: bnj1384 32201 |
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