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Theorem funcnv5mpt 30530
 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 30529 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
7 nne 2956 . . . . . . . . 9 𝐵𝐶𝐵 = 𝐶)
8 eqvincg 3560 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
95, 8syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
107, 9syl5bb 286 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
1110imbi1d 346 . . . . . . 7 ((𝜑𝑥𝐴) → ((¬ 𝐵𝐶𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
12 orcom 868 . . . . . . . 8 ((𝑥 = 𝑧𝐵𝐶) ↔ (𝐵𝐶𝑥 = 𝑧))
13 df-or 846 . . . . . . . 8 ((𝐵𝐶𝑥 = 𝑧) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
1412, 13bitri 278 . . . . . . 7 ((𝑥 = 𝑧𝐵𝐶) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
15 19.23v 1944 . . . . . . 7 (∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1611, 14, 153bitr4g 318 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
1716ralbidv 3127 . . . . 5 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
18 ralcom4 3163 . . . . 5 (∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1917, 18bitrdi 290 . . . 4 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
201, 19ralbida 3159 . . 3 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
21 nfcv 2920 . . . . . 6 𝑧𝐴
22 nfv 1916 . . . . . 6 𝑥 𝑦 = 𝐶
23 funcnv5mpt.1 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐶)
2423eqeq2d 2770 . . . . . 6 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝐶))
252, 21, 22, 24rmo4f 3650 . . . . 5 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2625albii 1822 . . . 4 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
27 ralcom4 3163 . . . 4 (∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2826, 27bitr4i 281 . . 3 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2920, 28bitr4di 293 . 2 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
306, 29bitr4d 285 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  ∀wal 1537   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2112  Ⅎwnfc 2900   ≠ wne 2952  ∀wral 3071  ∃*wrmo 3074   ↦ cmpt 5113  ◡ccnv 5524  Fun wfun 6330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344 This theorem is referenced by:  funcnv4mpt  30531
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