Theorem bnj60 35045

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 35043	. . . 4 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
5		eqid 2729	. . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom 𝑔 ∩ dom ℎ)
6	1, 2, 3, 5	bnj1311 35007	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
7	6	3expia 1121	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → (ℎ ∈ 𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))))
8	7	ralrimiv 3124	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
9	8	ralrimiva 3125	. . . 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 34814	. . . 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 6521	. . 3 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐶)
16	13, 15	sylibr 234	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
17	1, 2, 3, 14	bnj1498 35044	. 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
18	16, 17	bnj1422 34820	1 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  ⟨cop 4591  ∪ cuni 4867  dom cdm 5631   ↾ cres 5633  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499   predc-bnj14 34671   FrSe w-bnj15 34675

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-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-bnj17 34670  df-bnj14 34672  df-bnj13 34674  df-bnj15 34676  df-bnj18 34678  df-bnj19 34680

This theorem is referenced by:  bnj1501  35050  bnj1523  35054