Theorem bnj145OLD 30049

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6337.

Hypotheses

Ref	Expression
bnj145OLD.1	⊢ 𝐴 ∈ V
bnj145OLD.2	⊢ (𝐹‘𝐴) ∈ V

Assertion

Ref	Expression
bnj145OLD	⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Proof of Theorem bnj145OLD

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj142OLD 30048	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑢 ∈ 𝐹 → 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
2		df-fn 5807	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3		bnj145OLD.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	snid 4155	. . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴}
5		eleq2 2677	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴}))
6	4, 5	mpbiri 247	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
7	6	anim2i 591	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
8	2, 7	sylbi 206	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
9		funfvop 6237	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	8, 9	syl 17	. . . . . 6 ⊢ (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2676	. . . . . 6 ⊢ (𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑢 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 236	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑢 ∈ 𝐹))
13	1, 12	impbid 201	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
14	13	alrimiv 1842	. . 3 ⊢ (𝐹 Fn {𝐴} → ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
15		velsn 4141	. . . . 5 ⊢ (𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩)
16	15	bibi2i 326	. . . 4 ⊢ ((𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}) ↔ (𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
17	16	albii 1737	. . 3 ⊢ (∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}) ↔ ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
18	14, 17	sylibr 223	. 2 ⊢ (𝐹 Fn {𝐴} → ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
19		dfcleq 2604	. 2 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} ↔ ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
20	18, 19	sylibr 223	1 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by: (None)