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Mirrors > Home > MPE Home > Th. List > wfr1OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of wfr1 8282 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfr1OLD.1 | ⊢ 𝑅 We 𝐴 |
wfr1OLD.2 | ⊢ 𝑅 Se 𝐴 |
wfr1OLD.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr1OLD | ⊢ 𝐹 Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfr1OLD.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
2 | wfr1OLD.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
3 | wfr1OLD.3 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | 1, 2, 3 | wfrfunOLD 8266 | . 2 ⊢ Fun 𝐹 |
5 | eqid 2733 | . . 3 ⊢ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) | |
6 | 1, 2, 3, 5 | wfrlem16OLD 8271 | . 2 ⊢ dom 𝐹 = 𝐴 |
7 | df-fn 6500 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
8 | 4, 6, 7 | mpbir2an 710 | 1 ⊢ 𝐹 Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3909 {csn 4587 ⟨cop 4593 Se wse 5587 We wwe 5588 dom cdm 5634 ↾ cres 5636 Predcpred 6253 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 wrecscwrecs 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 |
This theorem is referenced by: (None) |
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