Theorem fwfffun 35787

Description: If 𝜓 is function-like over 𝐴, then 𝐹 matches its behavior. (Contributed by BTernaryTau, 28-Sep-2026.)

Hypotheses

Ref	Expression
fwfffun.1	⊢ Ⅎ𝑦𝜓
fwfffun.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
fwfffun.3	⊢ 𝐹 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)}

Assertion

Ref	Expression
fwfffun	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝜓(𝑥, 𝑦, 𝑧)   𝐴(𝑥)   𝐹(𝑥, 𝑦, 𝑧)

Proof of Theorem fwfffun

Step	Hyp	Ref	Expression
1		fwfffun.3	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)}
2	1	eleq2i 2856	. . 3 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑧⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)})
3		opabidw 5502	. . 3 ⊢ (⟨𝑥, 𝑧⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓))
4	2, 3	bitri 278	. 2 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓))
5		fwfffun.2	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
6		fwfffun.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
7	6	rmo2 3838	. . . . . 6 ⊢ (∃*𝑧 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
8	7	ralbii 3112	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑧 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
9	5, 8	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃*𝑧 ∈ 𝐴 𝜓)
10		funopab 6595	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)} ↔ ∀𝑥∃*𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓))
11	1	funeqi 6580	. . . . 5 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)})
12		2reu5lem2 3717	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑧 ∈ 𝐴 𝜓 ↔ ∀𝑥∃*𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓))
13	10, 11, 12	3bitr4i 306	. . . 4 ⊢ (Fun 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃*𝑧 ∈ 𝐴 𝜓)
14	9, 13	sylibr 237	. . 3 ⊢ (𝜑 → Fun 𝐹)
15		df-br 5107	. . . 4 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
16		funbrfv2b 6965	. . . 4 ⊢ (Fun 𝐹 → (𝑥𝐹𝑧 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))
17	15, 16	bitr3id 288	. . 3 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))
18	14, 17	syl 18	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))
19	4, 18	bitr3id 288	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2147  ∃*wmo 2565  ∀wral 3080  ∃*wrmo 3368  ⟨cop 4838   class class class wbr 5106  {copab 5170  dom cdm 5655  Fun wfun 6552  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566

This theorem is used by:  fwfffunres  35788