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Theorem funcnv5mpt 29799
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 29798 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
7 nne 2936 . . . . . . . . 9 𝐵𝐶𝐵 = 𝐶)
8 eqvincg 3468 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
95, 8syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
107, 9syl5bb 272 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
1110imbi1d 330 . . . . . . 7 ((𝜑𝑥𝐴) → ((¬ 𝐵𝐶𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
12 orcom 401 . . . . . . . 8 ((𝑥 = 𝑧𝐵𝐶) ↔ (𝐵𝐶𝑥 = 𝑧))
13 df-or 384 . . . . . . . 8 ((𝐵𝐶𝑥 = 𝑧) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
1412, 13bitri 264 . . . . . . 7 ((𝑥 = 𝑧𝐵𝐶) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
15 19.23v 2020 . . . . . . 7 (∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1611, 14, 153bitr4g 303 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
1716ralbidv 3124 . . . . 5 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
18 ralcom4 3364 . . . . 5 (∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1917, 18syl6bb 276 . . . 4 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
201, 19ralbida 3120 . . 3 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
21 nfcv 2902 . . . . . 6 𝑧𝐴
22 nfv 1992 . . . . . 6 𝑥 𝑦 = 𝐶
23 funcnv5mpt.1 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐶)
2423eqeq2d 2770 . . . . . 6 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝐶))
252, 21, 22, 24rmo4f 29666 . . . . 5 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2625albii 1896 . . . 4 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
27 ralcom4 3364 . . . 4 (∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2826, 27bitr4i 267 . . 3 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2920, 28syl6bbr 278 . 2 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
306, 29bitr4d 271 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1630   = wceq 1632  wex 1853  wnf 1857  wcel 2139  wnfc 2889  wne 2932  wral 3050  ∃*wrmo 3053  cmpt 4881  ccnv 5265  Fun wfun 6043
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  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-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  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-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  funcnv4mpt  29800
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