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Theorem islmim 19264
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 19225 . . 3 LMIso = (𝑎 ∈ LMod, 𝑏 ∈ LMod ↦ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
2 ovex 6841 . . . 4 (𝑎 LMHom 𝑏) ∈ V
32rabex 4964 . . 3 {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
4 oveq12 6822 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑎 LMHom 𝑏) = (𝑅 LMHom 𝑆))
5 fveq2 6352 . . . . . 6 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
6 islmim.b . . . . . 6 𝐵 = (Base‘𝑅)
75, 6syl6eqr 2812 . . . . 5 (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
8 fveq2 6352 . . . . . 6 (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
9 islmim.c . . . . . 6 𝐶 = (Base‘𝑆)
108, 9syl6eqr 2812 . . . . 5 (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
11 f1oeq23 6291 . . . . 5 (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
127, 10, 11syl2an 495 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
134, 12rabeqbidv 3335 . . 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 7044 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
15 df-3an 1074 . 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 6288 . . . . 5 (𝑐 = 𝐹 → (𝑐:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
1716elrab 3504 . . . 4 (𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶} ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
1817anbi2i 732 . . 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 19235 . . . . . 6 (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod)
20 lmhmlmod2 19234 . . . . . 6 (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑆 ∈ LMod)
2119, 20jca 555 . . . . 5 (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
2221adantr 472 . . . 4 ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
2322pm4.71ri 668 . . 3 ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
2418, 23bitr4i 267 . 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 286 1 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  Basecbs 16059  LModclmod 19065   LMHom clmhm 19221   LMIso clmim 19222
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-rab 3059  df-v 3342  df-sbc 3577  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-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-lmhm 19224  df-lmim 19225
This theorem is referenced by:  lmimf1o  19265  lmimlmhm  19266  islmim2  19268  indlcim  20381  lmimco  20385  pwssplit4  38161
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