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Theorem fsneqrn 40143
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 6265 . . . . . . 7 (𝐹 Fn 𝐵𝐹:𝐵⟶ran 𝐹)
31, 2sylib 210 . . . . . 6 (𝜑𝐹:𝐵⟶ran 𝐹)
4 fsneqrn.a . . . . . . . 8 (𝜑𝐴𝑉)
5 snidg 4396 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
64, 5syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
7 fsneqrn.b . . . . . . . . 9 𝐵 = {𝐴}
87a1i 11 . . . . . . . 8 (𝜑𝐵 = {𝐴})
98eqcomd 2803 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
106, 9eleqtrd 2878 . . . . . 6 (𝜑𝐴𝐵)
113, 10ffvelrnd 6584 . . . . 5 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
1211adantr 473 . . . 4 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐹)
13 simpr 478 . . . . 5 ((𝜑𝐹 = 𝐺) → 𝐹 = 𝐺)
1413rneqd 5554 . . . 4 ((𝜑𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
1512, 14eleqtrd 2878 . . 3 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
1615ex 402 . 2 (𝜑 → (𝐹 = 𝐺 → (𝐹𝐴) ∈ ran 𝐺))
17 simpr 478 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
18 fsneqrn.g . . . . . . . . . 10 (𝜑𝐺 Fn 𝐵)
19 dffn2 6256 . . . . . . . . . 10 (𝐺 Fn 𝐵𝐺:𝐵⟶V)
2018, 19sylib 210 . . . . . . . . 9 (𝜑𝐺:𝐵⟶V)
218feq2d 6240 . . . . . . . . 9 (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
2220, 21mpbid 224 . . . . . . . 8 (𝜑𝐺:{𝐴}⟶V)
234, 22rnsnf 40112 . . . . . . 7 (𝜑 → ran 𝐺 = {(𝐺𝐴)})
2423adantr 473 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺𝐴)})
2517, 24eleqtrd 2878 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ {(𝐺𝐴)})
26 elsni 4383 . . . . 5 ((𝐹𝐴) ∈ {(𝐺𝐴)} → (𝐹𝐴) = (𝐺𝐴))
2725, 26syl 17 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) = (𝐺𝐴))
284adantr 473 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐴𝑉)
291adantr 473 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
3018adantr 473 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
3128, 7, 29, 30fsneq 40138 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
3227, 31mpbird 249 . . 3 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
3332ex 402 . 2 (𝜑 → ((𝐹𝐴) ∈ ran 𝐺𝐹 = 𝐺))
3416, 33impbid 204 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3383  {csn 4366  ran crn 5311   Fn wfn 6094  wf 6095  cfv 6099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107
This theorem is referenced by:  ssmapsn  40148
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