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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1383 | 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 |
---|---|
bnj1383.1 | ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) |
bnj1383.2 | ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) |
bnj1383.3 | ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) |
Ref | Expression |
---|---|
bnj1383 | ⊢ (𝜓 → Fun ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1383.1 | . 2 ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) | |
2 | bnj1383.2 | . 2 ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) | |
3 | bnj1383.3 | . 2 ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) | |
4 | biid 263 | . 2 ⊢ ((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ↔ (𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴)) | |
5 | biid 263 | . 2 ⊢ (((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝑓) ↔ ((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝑓)) | |
6 | biid 263 | . 2 ⊢ ((((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝑓) ∧ 𝑔 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝑔) ↔ (((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝑓) ∧ 𝑔 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝑔)) | |
7 | 1, 2, 3, 4, 5, 6 | bnj1379 32104 | 1 ⊢ (𝜓 → Fun ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 〈cop 4575 ∪ cuni 4840 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: bnj1385 32106 bnj60 32336 |
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