Theorem bnj60 35220

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 35218	. . . 4 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
5		eqid 2737	. . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom 𝑔 ∩ dom ℎ)
6	1, 2, 3, 5	bnj1311 35182	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
7	6	3expia 1122	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → (ℎ ∈ 𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))))
8	7	ralrimiv 3128	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
9	8	ralrimiva 3129	. . . 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 34989	. . . 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 588	. . 3 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐶)
14		bnj60.4	. . . 4 ⊢ 𝐹 = ∪ 𝐶
15	14	funeqi 6514	. . 3 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐶)
16	13, 15	sylibr 234	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
17	1, 2, 3, 14	bnj1498 35219	. 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
18	16, 17	bnj1422 34995	1 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3061   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4587  ∪ cuni 4864  dom cdm 5625   ↾ cres 5627  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493   predc-bnj14 34846   FrSe w-bnj15 34850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7811  df-1o 8399  df-bnj17 34845  df-bnj14 34847  df-bnj13 34849  df-bnj15 34851  df-bnj18 34853  df-bnj19 34855

This theorem is referenced by:  bnj1501  35225  bnj1523  35229