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Theorem bnj145OLD 30500
 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 6386.
Hypotheses
Ref Expression
bnj145OLD.1 𝐴 ∈ V
bnj145OLD.2 (𝐹𝐴) ∈ V
Assertion
Ref Expression
bnj145OLD (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})

Proof of Theorem bnj145OLD
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 bnj142OLD 30499 . . . . 5 (𝐹 Fn {𝐴} → (𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))
2 df-fn 5850 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3 bnj145OLD.1 . . . . . . . . . . 11 𝐴 ∈ V
43snid 4179 . . . . . . . . . 10 𝐴 ∈ {𝐴}
5 eleq2 2687 . . . . . . . . . 10 (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹𝐴 ∈ {𝐴}))
64, 5mpbiri 248 . . . . . . . . 9 (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
76anim2i 592 . . . . . . . 8 ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹𝐴 ∈ dom 𝐹))
82, 7sylbi 207 . . . . . . 7 (𝐹 Fn {𝐴} → (Fun 𝐹𝐴 ∈ dom 𝐹))
9 funfvop 6285 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
108, 9syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2686 . . . . . 6 (𝑢 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑢𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 237 . . . . 5 (𝐹 Fn {𝐴} → (𝑢 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑢𝐹))
131, 12impbid 202 . . . 4 (𝐹 Fn {𝐴} → (𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))
1413alrimiv 1852 . . 3 (𝐹 Fn {𝐴} → ∀𝑢(𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))
15 velsn 4164 . . . . 5 (𝑢 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑢 = ⟨𝐴, (𝐹𝐴)⟩)
1615bibi2i 327 . . . 4 ((𝑢𝐹𝑢 ∈ {⟨𝐴, (𝐹𝐴)⟩}) ↔ (𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))
1716albii 1744 . . 3 (∀𝑢(𝑢𝐹𝑢 ∈ {⟨𝐴, (𝐹𝐴)⟩}) ↔ ∀𝑢(𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))
1814, 17sylibr 224 . 2 (𝐹 Fn {𝐴} → ∀𝑢(𝑢𝐹𝑢 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
19 dfcleq 2615 . 2 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} ↔ ∀𝑢(𝑢𝐹𝑢 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
2018, 19sylibr 224 1 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480   ∈ wcel 1987  Vcvv 3186  {csn 4148  ⟨cop 4154  dom cdm 5074  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855 This theorem is referenced by: (None)
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