wfrfun - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wfrfun		Structured version Visualization version GIF version

Theorem wfrfun 8341

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 5640	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 486	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5641	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5577	. . . 4 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 18	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 486	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 490	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8		wfrfun.1	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9		df-wrecs 8330	. . . 4 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
10	8, 9	eqtri 2783	. . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
11	10	fprfung 8327	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
12	2, 6, 7, 11	syl3anc 1398	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 Po wpo 5556 Or wor 5557 Fr wfr 5600 Se wse 5601 We wwe 5602 ∘ ccom 5654 Fun wfun 6532 2^nd c2nd 8000 frecscfrecs 8298 wrecscwrecs 8329

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-iota 6494 df-fun 6540 df-fn 6541 df-fv 6546 df-ov 7423 df-frecs 8299 df-wrecs 8330

This theorem is used by: bpolylem 16214

W3C validator