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Theorem fsneqrn 41339
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 6522 . . . . . . 7 (𝐹 Fn 𝐵𝐹:𝐵⟶ran 𝐹)
31, 2sylib 219 . . . . . 6 (𝜑𝐹:𝐵⟶ran 𝐹)
4 fsneqrn.a . . . . . . . 8 (𝜑𝐴𝑉)
5 snidg 4596 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
64, 5syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
7 fsneqrn.b . . . . . . . . 9 𝐵 = {𝐴}
87a1i 11 . . . . . . . 8 (𝜑𝐵 = {𝐴})
98eqcomd 2832 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
106, 9eleqtrd 2920 . . . . . 6 (𝜑𝐴𝐵)
113, 10ffvelrnd 6848 . . . . 5 (𝜑 → (𝐹𝐴) ∈ ran 𝐹)
1211adantr 481 . . . 4 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐹)
13 simpr 485 . . . . 5 ((𝜑𝐹 = 𝐺) → 𝐹 = 𝐺)
1413rneqd 5807 . . . 4 ((𝜑𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
1512, 14eleqtrd 2920 . . 3 ((𝜑𝐹 = 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
1615ex 413 . 2 (𝜑 → (𝐹 = 𝐺 → (𝐹𝐴) ∈ ran 𝐺))
17 simpr 485 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ ran 𝐺)
18 fsneqrn.g . . . . . . . . . 10 (𝜑𝐺 Fn 𝐵)
19 dffn2 6513 . . . . . . . . . 10 (𝐺 Fn 𝐵𝐺:𝐵⟶V)
2018, 19sylib 219 . . . . . . . . 9 (𝜑𝐺:𝐵⟶V)
218feq2d 6497 . . . . . . . . 9 (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
2220, 21mpbid 233 . . . . . . . 8 (𝜑𝐺:{𝐴}⟶V)
234, 22rnsnf 41309 . . . . . . 7 (𝜑 → ran 𝐺 = {(𝐺𝐴)})
2423adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺𝐴)})
2517, 24eleqtrd 2920 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) ∈ {(𝐺𝐴)})
26 elsni 4581 . . . . 5 ((𝐹𝐴) ∈ {(𝐺𝐴)} → (𝐹𝐴) = (𝐺𝐴))
2725, 26syl 17 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹𝐴) = (𝐺𝐴))
284adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐴𝑉)
291adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
3018adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
3128, 7, 29, 30fsneq 41334 . . . 4 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
3227, 31mpbird 258 . . 3 ((𝜑 ∧ (𝐹𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
3332ex 413 . 2 (𝜑 → ((𝐹𝐴) ∈ ran 𝐺𝐹 = 𝐺))
3416, 33impbid 213 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  {csn 4564  ran crn 5555   Fn wfn 6347  wf 6348  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360
This theorem is referenced by:  ssmapsn  41344
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