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Theorem islmim 18829
Description: An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b 𝐵 = (Base‘𝑅)
islmim.c 𝐶 = (Base‘𝑆)
Assertion
Ref Expression
islmim (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))

Proof of Theorem islmim
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmim 18790 . . 3 LMIso = (𝑎 ∈ LMod, 𝑏 ∈ LMod ↦ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
2 ovex 6555 . . . 4 (𝑎 LMHom 𝑏) ∈ V
32rabex 4735 . . 3 {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
4 oveq12 6536 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑎 LMHom 𝑏) = (𝑅 LMHom 𝑆))
5 fveq2 6088 . . . . . 6 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
6 islmim.b . . . . . 6 𝐵 = (Base‘𝑅)
75, 6syl6eqr 2661 . . . . 5 (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
8 fveq2 6088 . . . . . 6 (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
9 islmim.c . . . . . 6 𝐶 = (Base‘𝑆)
108, 9syl6eqr 2661 . . . . 5 (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
11 f1oeq23 6028 . . . . 5 (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
127, 10, 11syl2an 492 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
134, 12rabeqbidv 3167 . . 3 ((𝑎 = 𝑅𝑏 = 𝑆) → {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} = {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶})
141, 3, 13elovmpt2 6754 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
15 df-3an 1032 . 2 ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
16 f1oeq1 6025 . . . . 5 (𝑐 = 𝐹 → (𝑐:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
1716elrab 3330 . . . 4 (𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶} ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
1817anbi2i 725 . . 3 (((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
19 lmhmlmod1 18800 . . . . . 6 (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod)
20 lmhmlmod2 18799 . . . . . 6 (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑆 ∈ LMod)
2119, 20jca 552 . . . . 5 (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
2221adantr 479 . . . 4 ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
2322pm4.71ri 662 . . 3 ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
2418, 23bitr4i 265 . 2 (((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
2514, 15, 243bitri 284 1 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {crab 2899  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Basecbs 15641  LModclmod 18632   LMHom clmhm 18786   LMIso clmim 18787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-lmhm 18789  df-lmim 18790
This theorem is referenced by:  lmimf1o  18830  lmimlmhm  18831  islmim2  18833  indlcim  19940  lmimco  19944  pwssplit4  36473
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