Theorem isarep1 5416

Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
isarep1	⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑏,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑏)   𝐴(𝑦,𝑏)

Proof of Theorem isarep1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2805	. . 3 ⊢ 𝑏 ∈ V
2	1	elima 5081	. 2 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3		df-br 4089	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opelopabsb 4354	. . . 4 ⊢ (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5		sbsbc 3035	. . . . . 6 ⊢ ([𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)
6	5	sbbii 1813	. . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7		sbsbc 3035	. . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
8	6, 7	bitr2i 185	. . . 4 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
9	3, 4, 8	3bitri 206	. . 3 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
10	9	rexbii 2539	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11		nfs1v 1992	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12		nfv 1576	. . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑
13		sbequ12r 1820	. . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14	11, 12, 13	cbvrex 2764	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
15	2, 10, 14	3bitri 206	1 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1810   ∈ wcel 2202  ∃wrex 2511  [wsbc 3031  ⟨cop 3672   class class class wbr 4088  {copab 4149   “ cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)