Theorem bnj60 35054

Description: Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj60.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj60.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj60.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj60.4	⊢ 𝐹 = ∪ 𝐶

Assertion

Ref	Expression
bnj60	⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj60

Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj60.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj60.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj60.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4	1, 2, 3	bnj1497 35052	. . . 4 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
5		eqid 2734	. . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom 𝑔 ∩ dom ℎ)
6	1, 2, 3, 5	bnj1311 35016	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
7	6	3expia 1120	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → (ℎ ∈ 𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))))
8	7	ralrimiv 3142	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
9	8	ralrimiva 3143	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
10		biid 261	. . . . 5 ⊢ (∀𝑔 ∈ 𝐶 Fun 𝑔 ↔ ∀𝑔 ∈ 𝐶 Fun 𝑔)
11		biid 261	. . . . 5 ⊢ ((∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) ↔ (∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))))
12	10, 5, 11	bnj1383 34823	. . . 4 ⊢ ((∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) → Fun ∪ 𝐶)
13	4, 9, 12	sylancr 587	. . 3 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐶)
14		bnj60.4	. . . 4 ⊢ 𝐹 = ∪ 𝐶
15	14	funeqi 6588	. . 3 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐶)
16	13, 15	sylibr 234	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
17	1, 2, 3, 14	bnj1498 35053	. 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
18	16, 17	bnj1422 34829	1 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {cab 2711  ∀wral 3058  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  ⟨cop 4636  ∪ cuni 4911  dom cdm 5688   ↾ cres 5690  Fun wfun 6556   Fn wfn 6557  ‘cfv 6562   predc-bnj14 34680   FrSe w-bnj15 34684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-bnj17 34679  df-bnj14 34681  df-bnj13 34683  df-bnj15 34685  df-bnj18 34687  df-bnj19 34689

This theorem is referenced by:  bnj1501  35059  bnj1523  35063