Theorem bnj60 32942

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 32940	. . . 4 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
5		eqid 2738	. . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom 𝑔 ∩ dom ℎ)
6	1, 2, 3, 5	bnj1311 32904	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
7	6	3expia 1119	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → (ℎ ∈ 𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))))
8	7	ralrimiv 3106	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
9	8	ralrimiva 3107	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
10		biid 260	. . . . 5 ⊢ (∀𝑔 ∈ 𝐶 Fun 𝑔 ↔ ∀𝑔 ∈ 𝐶 Fun 𝑔)
11		biid 260	. . . . 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 32711	. . . 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 586	. . 3 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐶)
14		bnj60.4	. . . 4 ⊢ 𝐹 = ∪ 𝐶
15	14	funeqi 6439	. . 3 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐶)
16	13, 15	sylibr 233	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
17	1, 2, 3, 14	bnj1498 32941	. 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
18	16, 17	bnj1422 32717	1 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4564  ∪ cuni 4836  dom cdm 5580   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-bnj17 32566  df-bnj14 32568  df-bnj13 32570  df-bnj15 32572  df-bnj18 32574  df-bnj19 32576

This theorem is referenced by:  bnj1501  32947  bnj1523  32951