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Theorem funcnv5mpt 29939
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
Hypotheses
Ref Expression
funcnvmpt.0 𝑥𝜑
funcnvmpt.1 𝑥𝐴
funcnvmpt.2 𝑥𝐹
funcnvmpt.3 𝐹 = (𝑥𝐴𝐵)
funcnvmpt.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
funcnv5mpt.1 (𝑥 = 𝑧𝐵 = 𝐶)
Assertion
Ref Expression
funcnv5mpt (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
Distinct variable groups:   𝑥,𝑧   𝜑,𝑧   𝑧,𝐴   𝑧,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑧)   𝐹(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem funcnv5mpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funcnvmpt.0 . . 3 𝑥𝜑
2 funcnvmpt.1 . . 3 𝑥𝐴
3 funcnvmpt.2 . . 3 𝑥𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
61, 2, 3, 4, 5funcnvmpt 29938 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
7 nne 2941 . . . . . . . . 9 𝐵𝐶𝐵 = 𝐶)
8 eqvincg 3483 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
95, 8syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
107, 9syl5bb 274 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
1110imbi1d 332 . . . . . . 7 ((𝜑𝑥𝐴) → ((¬ 𝐵𝐶𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
12 orcom 896 . . . . . . . 8 ((𝑥 = 𝑧𝐵𝐶) ↔ (𝐵𝐶𝑥 = 𝑧))
13 df-or 874 . . . . . . . 8 ((𝐵𝐶𝑥 = 𝑧) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
1412, 13bitri 266 . . . . . . 7 ((𝑥 = 𝑧𝐵𝐶) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
15 19.23v 2037 . . . . . . 7 (∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1611, 14, 153bitr4g 305 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
1716ralbidv 3133 . . . . 5 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
18 ralcom4 3377 . . . . 5 (∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1917, 18syl6bb 278 . . . 4 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
201, 19ralbida 3129 . . 3 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
21 nfcv 2907 . . . . . 6 𝑧𝐴
22 nfv 2009 . . . . . 6 𝑥 𝑦 = 𝐶
23 funcnv5mpt.1 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐶)
2423eqeq2d 2775 . . . . . 6 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝐶))
252, 21, 22, 24rmo4f 3565 . . . . 5 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2625albii 1914 . . . 4 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
27 ralcom4 3377 . . . 4 (∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2826, 27bitr4i 269 . . 3 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2920, 28syl6bbr 280 . 2 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
306, 29bitr4d 273 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  wal 1650   = wceq 1652  wex 1874  wnf 1878  wcel 2155  wnfc 2894  wne 2937  wral 3055  ∃*wrmo 3058  cmpt 4890  ccnv 5278  Fun wfun 6064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078
This theorem is referenced by:  funcnv4mpt  29940
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