wfrfun - Metamath Proof Explorer

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Theorem wfrfun 8302

Description: The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.)

**Hypothesis**
Ref	Expression
wfrfun.1	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

**Assertion**
Ref	Expression
wfrfun	⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

**Proof of Theorem wfrfun**
Step	Hyp	Ref	Expression
1		wefr 5628	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 480	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5629	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5565	. . . 4 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 17	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 480	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 484	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8		wfrfun.1	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9		df-wrecs 8291	. . . 4 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
10	8, 9	eqtri 2752	. . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
11	10	fprfung 8288	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
12	2, 6, 7, 11	syl3anc 1373	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Po wpo 5544 Or wor 5545 Fr wfr 5588 Se wse 5589 We wwe 5590 ∘ ccom 5642 Fun wfun 6505 2^nd c2nd 7967 frecscfrecs 8259 wrecscwrecs 8290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-iota 6464 df-fun 6513 df-fn 6514 df-fv 6519 df-ov 7390 df-frecs 8260 df-wrecs 8291

This theorem is referenced by: bpolylem 16014

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