Theorem bnj145OLD 30795

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 6432.

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 30794	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑢 ∈ 𝐹 → 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
2		df-fn 5891	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3		bnj145OLD.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	snid 4208	. . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴}
5		eleq2 2690	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴}))
6	4, 5	mpbiri 248	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
7	6	anim2i 593	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
8	2, 7	sylbi 207	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
9		funfvop 6329	. . . . . . 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 2689	. . . . . 6 ⊢ (𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑢 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 237	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑢 ∈ 𝐹))
13	1, 12	impbid 202	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
14	13	alrimiv 1855	. . 3 ⊢ (𝐹 Fn {𝐴} → ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
15		velsn 4193	. . . . 5 ⊢ (𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩)
16	15	bibi2i 327	. . . 4 ⊢ ((𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}) ↔ (𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
17	16	albii 1747	. . 3 ⊢ (∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}) ↔ ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 = ⟨𝐴, (𝐹‘𝐴)⟩))
18	14, 17	sylibr 224	. 2 ⊢ (𝐹 Fn {𝐴} → ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
19		dfcleq 2616	. 2 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} ↔ ∀𝑢(𝑢 ∈ 𝐹 ↔ 𝑢 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
20	18, 19	sylibr 224	1 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183  dom cdm 5114  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)