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Theorem funcnv5mpt 30103
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 30102 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
7 nne 2988 . . . . . . . . 9 𝐵𝐶𝐵 = 𝐶)
8 eqvincg 3580 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
95, 8syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
107, 9syl5bb 284 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦 = 𝐶)))
1110imbi1d 343 . . . . . . 7 ((𝜑𝑥𝐴) → ((¬ 𝐵𝐶𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
12 orcom 865 . . . . . . . 8 ((𝑥 = 𝑧𝐵𝐶) ↔ (𝐵𝐶𝑥 = 𝑧))
13 df-or 843 . . . . . . . 8 ((𝐵𝐶𝑥 = 𝑧) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
1412, 13bitri 276 . . . . . . 7 ((𝑥 = 𝑧𝐵𝐶) ↔ (¬ 𝐵𝐶𝑥 = 𝑧))
15 19.23v 1920 . . . . . . 7 (∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1611, 14, 153bitr4g 315 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
1716ralbidv 3164 . . . . 5 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
18 ralcom4 3199 . . . . 5 (∀𝑧𝐴𝑦((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
1917, 18syl6bb 288 . . . 4 ((𝜑𝑥𝐴) → (∀𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
201, 19ralbida 3194 . . 3 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧)))
21 nfcv 2949 . . . . . 6 𝑧𝐴
22 nfv 1892 . . . . . 6 𝑥 𝑦 = 𝐶
23 funcnv5mpt.1 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐶)
2423eqeq2d 2805 . . . . . 6 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝐶))
252, 21, 22, 24rmo4f 3660 . . . . 5 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2625albii 1801 . . . 4 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
27 ralcom4 3199 . . . 4 (∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦𝑥𝐴𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2826, 27bitr4i 279 . . 3 (∀𝑦∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∀𝑥𝐴𝑦𝑧𝐴 ((𝑦 = 𝐵𝑦 = 𝐶) → 𝑥 = 𝑧))
2920, 28syl6bbr 290 . 2 (𝜑 → (∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶) ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
306, 29bitr4d 283 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842  wal 1520   = wceq 1522  wex 1761  wnf 1765  wcel 2081  wnfc 2933  wne 2984  wral 3105  ∃*wrmo 3108  cmpt 5041  ccnv 5442  Fun wfun 6219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233
This theorem is referenced by:  funcnv4mpt  30104
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