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Theorem fsneqrn 45216
Description: Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fsneqrn.a (𝜑𝐴𝑉)
fsneqrn.b 𝐵 = {𝐴}
fsneqrn.f (𝜑𝐹 Fn 𝐵)
fsneqrn.g (𝜑𝐺 Fn 𝐵)
Assertion
Ref Expression
fsneqrn (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))

Proof of Theorem fsneqrn
StepHypRef Expression
1 fsneqrn.f . . . . . . 7 (𝜑𝐹 Fn 𝐵)
2 dffn3 6748 . . . . . . 7 (𝐹 Fn 𝐵𝐹:𝐵⟶ran 𝐹)
31, 2sylib 218 . . . . . 6 (𝜑𝐹:𝐵⟶ran 𝐹)
4 fsneqrn.a . . . . . . . 8 (𝜑𝐴𝑉)
5 snidg 4660 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
64, 5syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
7 fsneqrn.b . . . . . . . . 9 𝐵 = {𝐴}
87a1i 11 . . . . . . . 8 (𝜑𝐵 = {𝐴})
98eqcomd 2743 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
106, 9eleqtrd 2843 . . . . . 6 (𝜑𝐴𝐵)
113, 10ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
1211adantr 480 . . . 4 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐹)
13 simpr 484 . . . . 5 ((𝜑𝐹 = 𝐺) → 𝐹 = 𝐺)
1413rneqd 5949 . . . 4 ((𝜑𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
1512, 14eleqtrd 2843 . . 3 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
1615ex 412 . 2 (𝜑 → (𝐹 = 𝐺 → (𝐹𝐴) ∈ ran 𝐺))
17 simpr 484 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
18 fsneqrn.g . . . . . . . . . 10 (𝜑𝐺 Fn 𝐵)
19 dffn2 6738 . . . . . . . . . 10 (𝐺 Fn 𝐵𝐺:𝐵⟶V)
2018, 19sylib 218 . . . . . . . . 9 (𝜑𝐺:𝐵⟶V)
218feq2d 6722 . . . . . . . . 9 (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
2220, 21mpbid 232 . . . . . . . 8 (𝜑𝐺:{𝐴}⟶V)
234, 22rnsnf 45189 . . . . . . 7 (𝜑 → ran 𝐺 = {(𝐺𝐴)})
2423adantr 480 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺𝐴)})
2517, 24eleqtrd 2843 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ {(𝐺𝐴)})
26 elsni 4643 . . . . 5 ((𝐹𝐴) ∈ {(𝐺𝐴)} → (𝐹𝐴) = (𝐺𝐴))
2725, 26syl 17 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) = (𝐺𝐴))
284adantr 480 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐴𝑉)
291adantr 480 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
3018adantr 480 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
3128, 7, 29, 30fsneq 45211 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
3227, 31mpbird 257 . . 3 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
3332ex 412 . 2 (𝜑 → ((𝐹𝐴) ∈ ran 𝐺𝐹 = 𝐺))
3416, 33impbid 212 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  ssmapsn  45221
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