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Mathbox for Jonathan Ben-Naim |
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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 264 | . 2 ⊢ ((𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴) ↔ (𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴)) | |
5 | biid 264 | . 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 264 | . 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 32212 | 1 ⊢ (𝜓 → Fun ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 〈cop 4531 ∪ cuni 4800 dom cdm 5519 ↾ cres 5521 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: bnj1385 32214 bnj60 32444 |
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