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Theorem fsneqrn 39902
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 6215 . . . . . . 7 (𝐹 Fn 𝐵𝐹:𝐵⟶ran 𝐹)
31, 2sylib 208 . . . . . 6 (𝜑𝐹:𝐵⟶ran 𝐹)
4 fsneqrn.a . . . . . . . 8 (𝜑𝐴𝑉)
5 snidg 4351 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
64, 5syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
7 fsneqrn.b . . . . . . . . 9 𝐵 = {𝐴}
87a1i 11 . . . . . . . 8 (𝜑𝐵 = {𝐴})
98eqcomd 2766 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
106, 9eleqtrd 2841 . . . . . 6 (𝜑𝐴𝐵)
113, 10ffvelrnd 6523 . . . . 5 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
1211adantr 472 . . . 4 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐹)
13 simpr 479 . . . . 5 ((𝜑𝐹 = 𝐺) → 𝐹 = 𝐺)
1413rneqd 5508 . . . 4 ((𝜑𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
1512, 14eleqtrd 2841 . . 3 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
1615ex 449 . 2 (𝜑 → (𝐹 = 𝐺 → (𝐹𝐴) ∈ ran 𝐺))
17 simpr 479 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
18 fsneqrn.g . . . . . . . . . 10 (𝜑𝐺 Fn 𝐵)
19 dffn2 6208 . . . . . . . . . 10 (𝐺 Fn 𝐵𝐺:𝐵⟶V)
2018, 19sylib 208 . . . . . . . . 9 (𝜑𝐺:𝐵⟶V)
218feq2d 6192 . . . . . . . . 9 (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
2220, 21mpbid 222 . . . . . . . 8 (𝜑𝐺:{𝐴}⟶V)
234, 22rnsnf 39869 . . . . . . 7 (𝜑 → ran 𝐺 = {(𝐺𝐴)})
2423adantr 472 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺𝐴)})
2517, 24eleqtrd 2841 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ {(𝐺𝐴)})
26 elsni 4338 . . . . 5 ((𝐹𝐴) ∈ {(𝐺𝐴)} → (𝐹𝐴) = (𝐺𝐴))
2725, 26syl 17 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) = (𝐺𝐴))
284adantr 472 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐴𝑉)
291adantr 472 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
3018adantr 472 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
3128, 7, 29, 30fsneq 39897 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
3227, 31mpbird 247 . . 3 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
3332ex 449 . 2 (𝜑 → ((𝐹𝐴) ∈ ran 𝐺𝐹 = 𝐺))
3416, 33impbid 202 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  ssmapsn  39907
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