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Mirrors > Home > MPE Home > Th. List > xpscf | Structured version Visualization version GIF version |
Description: Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpscf | ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifid 4571 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2831 | . . . . 5 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 3091 | . . . 4 ⊢ (∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
4 | 3 | anbi2i 623 | . . 3 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) |
5 | df-3an 1088 | . . . 4 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) | |
6 | elixp2 8940 | . . . 4 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) | |
7 | 2onn 8679 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | fnex 7237 | . . . . . . 7 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ 2o ∈ ω) → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) | |
9 | 7, 8 | mpan2 691 | . . . . . 6 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) |
10 | 9 | pm4.71ri 560 | . . . . 5 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o)) |
11 | 10 | anbi1i 624 | . . . 4 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) |
12 | 5, 6, 11 | 3bitr4i 303 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) |
13 | ffnfv 7139 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) | |
14 | 4, 12, 13 | 3bitr4i 303 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ {〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴) |
15 | xpsfrnel2 17611 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | |
16 | 14, 15 | bitr3i 277 | 1 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∅c0 4339 ifcif 4531 {cpr 4633 〈cop 4637 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 ωcom 7887 1oc1o 8498 2oc2o 8499 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-2o 8506 df-ixp 8937 df-en 8985 df-fin 8988 |
This theorem is referenced by: xpsmnd 18803 xpsgrp 19090 dmdprdpr 20084 dprdpr 20085 xpsrngd 20197 xpsringd 20346 xpstopnlem1 23833 xpstps 23834 xpsxms 24563 xpsms 24564 |
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