Theorem bnj1500 34732

Description: Well-founded recursion, part 2 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
bnj1500.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1500.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1500.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1500.4	⊢ 𝐹 = ∪ 𝐶

Assertion

Ref	Expression
bnj1500	⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

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

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

Proof of Theorem bnj1500

Step	Hyp	Ref	Expression
1		bnj1500.1	. 2 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1500.2	. 2 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1500.3	. 2 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		bnj1500.4	. 2 ⊢ 𝐹 = ∪ 𝐶
5		biid 260	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))
6		biid 260	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
7		biid 260	. 2 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))
8	1, 2, 3, 4, 5, 6, 7	bnj1501 34731	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2705  ∀wral 3058  ∃wrex 3067   ⊆ wss 3949  ⟨cop 4638  ∪ cuni 4912  dom cdm 5682   ↾ cres 5684   Fn wfn 6548  ‘cfv 6553   predc-bnj14 34352   FrSe w-bnj15 34356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-bnj17 34351  df-bnj14 34353  df-bnj13 34355  df-bnj15 34357  df-bnj18 34359  df-bnj19 34361

This theorem is referenced by:  bnj1523  34735