Theorem isarep1 6575

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.) (Proof shortened by SN, 19-Dec-2024.)

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 3435	. . 3 ⊢ 𝑏 ∈ V
2	1	elima 6018	. 2 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3		df-br 5074	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		vopelopabsb 5472	. . . 4 ⊢ (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5	3, 4	bitri 276	. . 3 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
6	5	rexbii 3086	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7		nfs1v 2167	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
8		nfv 1921	. . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑
9		sbequ12r 2264	. . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
10	7, 8, 9	cbvrexw 3282	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
11	2, 6, 10	3bitri 298	1 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

Syntax hints:   ↔ wb 207  [wsb 2073   ∈ wcel 2119  ∃wrex 3063  ⟨cop 4562   class class class wbr 5073  {copab 5135   “ cima 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632

This theorem is referenced by: (None)